## Abstract

In this paper, we study modules having only finitely many submodules over any ring which is not necessarily commutative. We try to understand how such a module decomposes as a direct sum. We justify that any module V having only finitely many submodules over any ring A is an extension of a cyclic A-module by a finite A-module. Under some assumptions on A, such as commutativity of A, we prove that an A-module V has finitely many submodules if and only if V can be written as a direct sum of a cyclic A-module having only finitely many A-submodules and a finite A-module.

Original language | English |
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Pages (from-to) | 463-468 |

Number of pages | 6 |

Journal | Algebra Colloquium |

Volume | 23 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Sept 2016 |

### Bibliographical note

Publisher Copyright:© 2016 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.

### Funding

The first author is indebted to School of Mathematics, Institute for Research in Funda-mental Sciences (IPM) for support. The research of the first author was supported in part by a grant from IPM (No. 93050212).

Funders | Funder number |
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Institute for Research in Funda-mental Sciences | |

Institute for Research in Fundamental Sciences |

## Keywords

- direct sum
- module
- submodule